The Road to Reality: A Complete Guide to the Laws of the Universe
Jonathan Cape, London, 2004
“What laws govern our Universe?” is the opening question of Penrose’s magisterial and encyclopaedic account of the present state of Mathematics and Physics as applied to understanding the Universe and the laws that govern it. The question is not, of course, answered in this book any more than it has been answered at any time over the last two and a half thousand years, but it is still a question worth asking.
The conception behind Penrose’s book is quixotic in its dream of engaging a readership that has no specialist knowledge, and serving them mathematical formulae by the shed-load. Like Don Quixote, the project sort of succeeds, but not quite in the way expected.
The book divides itself into roughly two parts with the more basic Mathematics covering the first sixteen chapters, with the second half of the book devoted to “the actual pictures of the physical world that theory and observation have led us into”. The quotation is Penrose’s and neatly sums up his philosophy. In fact the book is dominated by pictures. For Penrose the world of Mathematics is visual; he can “see” the equations because he has their pictures in his mind’s eye and he shares with us these images throughout his book (and beyond - he has four drawings related to the book on show at the 2004 Royal Academy Summer Exhibition). The visual imagery in the book is a tremendous aid to understanding for readers who react better to pictures than equations, although I do have a couple of quibbles. First, some of the drawings are a bit small. I appreciate space is limited, but at 1094 pages perhaps some of the drawings could have been a little bigger; many of them have fine detail, which is sometimes quite difficult to see. I accept that my eyes are not what they were thirty years ago, but I bet the same applies to Penrose. A second point is that some of the graph diagrams do not show a scale: figures 4.1(p78) and 31.6 (p886) for instance.
My main criticism of the book is that the presentation of the material is often in the direction of the general to the specific, or from the more abstract to the less abstract. Even though it is a question of style, or more accurately tactics, in a book of this kind, the main preoccupation should be how the author can carry his readership along as far as possible. For example, there are two sections (10.3 and 10.4, p185-193), which lead from the notion of a dot product of two vectors (easy – you learn this at school in A level Maths) to the gradient of a vector field (first year university level). Nothing unusual here, except that Penrose does it the other way round. He starts with the geometric notion of a 1-form, moves to a vector field and ends up with a dot product. By introducing concepts in the direction “hard” to “easy”, you lose the chance of engaging the reader in the first place. A more extreme example is “Maxwell’s equations” (section 19.2 p442). The section opens with the question: “What, then are the Maxwell equations?” Penrose immediately jumps into the signature, the Maxwell field tensor and 2-forms, followed by the Hodge duals, the Levi-Civita tensor and bivectors. He then finishes with the orthogonal complement of the 2-plane element against a background of diagrams depicting the Hodge duals and Maxwell equations, which he writes down “very simply” as dF = 0, d*F=4π*J, - without even a passing reference to a curl. I am not saying that the formalism that Penrose is introducing to recast Maxwell’s equations is not important or useful, just that to engage a readership, who are mostly not specialists in the field, but are used to Maxwell’s equations in curl, div and grad form, surely you have to start from the easy and familiar and work your way, together with the reader, to the more difficult and unfamiliar areas.
Penrose’s main area of interest is in twistor theory. It is a shame that he doesn’t really introduce twistors until page 962. It is always best to write about what you know and feel passionate about, and the sections on twistors could easily have been much longer. Of course, this is a game one could always play with any encyclopaedic work: what extra should you put in?, what should you leave out? My personal list of extras would be: more on Category theory – particularly as it is an area of mathematics which is so full of imagery, Hopf algebra (quantum groups), and philosophy. Penrose does, to be fair, include quite a lot of a reflective nature, particularly, as the title of the book indicates, to do with the reality of mathematics and physics. But however important they are, there is more to philosophy than just Popper and Plato.
There is a bibliography and index at the end of the book and a section on notation at the beginning, with notes at the end of each chapter. A glossary of terms would have been useful, but more importantly there are no explicit “further reading” lists. I say “explicit” because in many of the chapter notes there are references to further reading, usually of the “see, for example” kind. Unless the reader is a specialist in all the areas covered by the book, a reading list of basic introductory works for each chapter would be an enormous aid.
I thought the section on fashion in physical theory quite superb (34.3 p1017). Everybody talks about beauty and order and even unity, but very few mention fashion. There is a bandwagon effect in the physical sciences, as in all other human endeavours, and this effect distorts scientist’s attitudes. Throughout the book one feels strongly the sense of passion that Penrose has for his subject, and in speculating about the next advances in quantum gravity, such passion, I am sure, impels him to plump for his own area of twistor theory. There is nothing wrong with passion in science; in fact, I don’t think that science could exist without it. I said at the beginning of this review that I felt Penrose succeeds, but not quite in the way expected. “The Road to Reality” represents a voice to be heard for mathematics and physics. In a world where the number of students studying these subjects at university is dramatically falling, and there is a general dumbing down of science, Penrose stands up for high scientific ideals and the importance and passion of understanding the fundamentals of our world. Maybe, the windmills are real after all.